[seqfan] Re: Farey series A006842/A006843

[Olivier Gerard]

On Sun, Apr 26, 2009 at 01:02, Peter Luschny
<peter.luschny at googlemail.com> wrote:
> The following nice non-recursive way
> to compute the Farey Tree has been
> discussed by Norman Routledge,
> "Computing Farey Series," The Mathematical
> Gazette, Vol. 29 (No. 523), 55–62 (March 2008).

It is not recursive on the previous set but it is recursive internally.

It maintains two steps of two associated sequences (4 values) and to
know when to stop it is computing implicitely  sum( eulerphi(i),  i=1..n )

>
> Perhaps this Maple code should be added to
> A006842/A006843 respectively?
>
> --

Here is one possible Mathematica Code

FareySequence[n_] :=
 Map[First,
  NestWhileList[{#1[[2]], #1[[2]]*
       Quotient[n + #1[[1, 2]], #[[2, 2]]] - #1[[1]]} &, {{0, 1}, {1, n}},
 #1[[2, 1]] < n & ]]

FareyNumerators[n_] := Map[First, FareySequence[n]]

FareyDenominators[n_] := Map[Last, FareySequence[n]]

> --
>
Olivier